Power tariffs reduce PR5901’s exponential invoice layers to one optimum

docs/ci-verification.md

@@ -1,6 +1,23 @@
 # CI verification
 
-## Current selected-family replay: #5599
+## Current selected-family replay: #6030
+
+The [selected record](../certificates/selected-result.json) binds conditional
+κ = 0.003125804 to the [proof review](research/pr-intake-evidence/pr6030-audit/PROOF-REVIEW.md),
+[validation receipts](research/pr-intake-evidence/pr6030-audit/VALIDATION.json),
+and [explicit source closure](../certificates/frontier6030-source.json).
+`make frontier6030-verify` follows that source-bound selected replay. The review
+separates retained normal primary macOS/Linux evidence from fresh independent
+word/invoice/native, original caller, adapted caller and optimized primary checks.
+Consult its receipts for exact commands, source heads and limits; the #5599
+checkpoint below retains its own historical replay details.
+
+The five inherited framework assumptions remain. Neither the full uniform word
+nor a complete rounded multiplier is materialized. A source/metadata check is
+not an operator replay or a formal theorem proof. Routine research submissions
+do not need to rerun this selected-family replay.
+
+## Historical selected-family replay: #5599
 
 The [selected record](../certificates/selected-result.json) binds conditional
 κ = 0.00310045 to the [proof review](research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md)

docs/research/current-status.md

@@ -1,6 +1,6 @@
 # Current reviewed status
 
-Selected conditional κ: **0.003125804 = 781451/250000000**, DreamingOfClouds'
+Selected conditional κ: **0.003125804** = 781451/250000000, DreamingOfClouds'
 PR6030 on the cumulative P104 construction. The complete caller counts 13,444
 productive fields and 66,834 paid calls. The saving is about 0.81775% above the
 preceding selected PR5599 claim 0.00310045.

research/power-layer-collapse/README.md

@@ -0,0 +1,126 @@
+# Power tariffs collapse the fixed-spread antichain to one optimum
+
+For every integer m >= 1 and 0 <= k <= m, the complete k-spread layer of
+[Nick Petalas's PR #5901](https://github.com/CrocSwap/integer-mult-bounds/pull/5901)
+`concave-invoice-antichains` family has **one unique cheapest invoice for every
+power tariff r^p with 0 < p < 1**: spread precisely blocks 1,...,k.
+The result answers that package's second open question. Its 252-member m=10
+layer and 462-member m=11 layer each reduce to one representative under this
+narrower tariff family, although every pair remains incomparable under the
+larger increasing-concave family.
+
+This is a scoped proof and exact finite corroboration, not a legal supplier,
+a new kappa, a claim about arbitrary invoices, or a refutation of the predecessor.
+The predecessor's all-concave theorem remains valid.
+
+## Contract and proof
+
+Write H_r for one rank-r paid call. For S subset {1,...,m}, retain the full bill
+
+```
+P_S = sum_(i not in S) 2 H_(2i) + sum_(i in S) (H_(2i-1)+H_(2i+1)),
+I_S = P_S - H_M,                 M = 2m(m+1).
+```
+
+All candidates have 2m paid calls, mass M, and the same productive target.
+No other fee, boundary, or consumer may differ in this comparison. The theorem
+is about the synthetic family defined above; it grants no realization of it.
+
+Fix p in (0,1). The cost difference made by spreading block i is
+
+```
+delta_i(p) = (2i-1)^p + (2i+1)^p - 2(2i)^p.
+```
+
+For x>1, integrating the second derivative twice gives
+
+```
+delta(x) = p(p-1) integral_[-1,1] (1-|t|) (x+t)^(p-2) dt.
+```
+
+The identity follows by applying the fundamental theorem of calculus on each
+side of x, or by integration by parts on [-1,0] and [0,1]. The kernel is
+nonnegative and positive in the interior. Since p-2<0, its power factor strictly
+decreases as x increases. Since p(p-1)<0, **delta(x) strictly increases**.
+Thus delta_1(p) < ... < delta_m(p) < 0, simultaneously for every p in (0,1).
+
+Let S0={1,...,k}. If a k-subset S differs from S0, it contains j>k and omits
+i<=k. Exchanging j for i decreases its complete bill by
+`delta_j(p)-delta_i(p)>0`. Repeating terminates at S0. The target and all
+unchanged paid blocks cancel at each exchange. This proves strict simultaneous
+dominance and uniqueness for 0<k<m; at k=0,m the layer is a singleton. A common
+normalization by any positive width to the power p preserves the result.
+
+The same argument applies to any increasing sequence of block centers x_i>1
+with a common symmetric spread h satisfying 0<h<min x_i: use
+`(x_i-h)^p+(x_i+h)^p-2*x_i^p` and the triangular kernel on [-h,h].
+It does not automatically cover varying spread widths.
+
+## Why the larger antichain is consistent
+
+At m=2 the bills for S={1} and S={2} differ by -1 under min(r,2) and by +1
+under min(r,4). These hinge witnesses prove their concave incomparability, as
+in the predecessor. Neither is a power tariff. In contrast, S={1} is strictly
+cheaper for every p in (0,1). Equal count and mass alone would not establish
+this; the ordered disjoint equal-width spreads supply the missing structure.
+
+A reducer can discard the rest of the **complete layer** if it is pricing only
+these powers. If a candidate pool omits S0, this theorem cannot manufacture a
+replacement. Changed physical boundaries, productive targets, fees, or legal
+realizability require separate checks. At p=0 or p=1 every layer member ties;
+the strict theorem deliberately excludes those endpoints. For p=3/2 the
+ordering reverses, so extending this pruning rule outside its interval is wrong.
+
+## Evidence and reproduction
+
+```
+python3.12 -B scripts/research_package.py --package research/power-layer-collapse
+python3.12 -B scripts/check_research_commit.py --package research/power-layer-collapse
+```
+
+The standard-library verifier enumerates every layer for 1<=m<=11. It evaluates
+all distinct rational p=a/b in (0,1), b<=8, using rational intervals obtained
+by **integer nth-root bounds**, never floating point. It checks every adjacent
+spread inequality and strict complete-invoice dominance of every noncanonical
+layer member. Endpoints, both hinge witnesses, the reversed p=3/2 order and
+changed-target/fee counterexamples are exact controls. Normal and optimized
+runs are registered; no package file is written by replay. The uniform theorem
+is the written proof, not extrapolation from these checks.
+
+Source provenance is pinned in SOURCES.json. Upstream was refreshed to
+721c03822c421b7f7fcbb2598993b947c4d88419 (revision 8) on 2026-10-28 UTC;
+the selected record there is #6030, kappa .003125804. The sparse baseline
+preflight reported 1,393 omitted inputs; that unrelated selected supplier replay
+was not materialized or run. Only the predecessor definition is a live input. The optional source-history checker
+refuses this shallow repository; no full-history pass is claimed. All pins resolve
+at the immediate main-branch base, which is an ancestor of the publication head.
+The ordinary committed-byte preflight passes.
+
+## Significance for researchers
+
+An exponential frontier for every concave tariff need not obstruct optimization
+of the power moments actually used for kappa. This family permits an exact
+one-representative reduction across the entire relevant power interval. The
+next useful question is whether legal supplier exchanges share its ordered
+spread structure, with all physical costs retained.
+
+## Open next questions
+
+Can a legal source-bound supplier realize two independently selectable equal-width
+spreads with a common complete boundary and fee? First extract two actual paid
+rank exchanges and their target, restoration and native ledgers. No such physical
+realization is supplied here; the synthetic family alone cannot raise kappa.
+
+For a partial layer that omits S0, can one characterize all power-undominated
+members? Start with the first index-incomparable pair and certify its difference
+on 0<p<1. The exchange proof only orders componentwise comparable index lists.
+
+## Credit
+
+Prepared by [zxv](https://github.com/zxv) with substantial OpenAI Codex assistance.
+[Nick Petalas](https://github.com/petalas) supplies the antichain family and open
+question. Its lineage credits [Ali Romman](https://github.com/aliromman),
+[zxv](https://github.com/zxv), [mightyOil](https://github.com/mightyOil) and [Dugongue](https://github.com/Dugongue) for the existing invoice
+framework; no predecessor construction is claimed as new. The triangular-kernel
+calculus identity and exchange argument are elementary. No independent human or
+proof-assistant review is claimed. Apache-2.0.

research/power-layer-collapse/SOURCES.json

@@ -0,0 +1,24 @@
+{
+  "schema_version": 1,
+  "base_commit": "721c03822c421b7f7fcbb2598993b947c4d88419",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "d59b5035e7b4cbaa9d688a1d32dd0d6eb79c466a",
+      "sha256": "9b8d0eea95b3bf0dfcb7938a732a6439b87a2685dec2a5ab9e525b3b34debbfa"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81"
+    },
+    {
+      "path": "research/concave-invoice-antichains/README.md",
+      "usage": "input",
+      "git_blob": "3fb016a040435cd437cc1905ba765b4e5ac18ed7",
+      "sha256": "d244d9f3ab98b0eb9101bb4e8a1305aca3b4299d249cb21757acbcd43e291279"
+    }
+  ]
+}

research/power-layer-collapse/TASK.json

@@ -0,0 +1,79 @@
+{
+  "schema_version": 1,
+  "task_id": "coverage-law-change",
+  "priority_revision": 8,
+  "base_commit": "721c03822c421b7f7fcbb2598993b947c4d88419",
+  "original_card": {
+    "id": "coverage-law-change",
+    "question": "Can a completed call earn more independent productive target credit without an offsetting paid bill?",
+    "starting_points": [
+      "docs/research/current-priorities.md",
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md"
+    ],
+    "methods": [
+      "Construct and independently check one source-bound instance of the stated first experiment.",
+      "Seek an exact counterexample or target-specific complete-budget obstruction to that same proposed change."
+    ],
+    "acceptance_gate": "A source-bound arbitrary-input identity and legal producer/consumer boundary with the complete paid and productive target invoice, including restoration, ownership, native alignment and extra charges. Price under the actual current model; historical ceiling.py assumptions must be established before use. A favorable optimistic relaxation keeps a route open but is not a completed supplier. A scoped local result remains useful; expansion requires the family-level budget gate.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_selected_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read revision 8, the selected review and live exact-source handoffs. Then follow this card's bounded first_experiment.",
+      "scope": "Source-bound orientation only, not a supplier replay or theorem proof."
+    },
+    "first_experiment": "Use #5599 as a completed positive control, then choose one uncovered cut/module beyond an applicable frozen exclusion. Derive independence, saturation and chronological availability rather than counting rational rank alone. Price target credits and all new jobs together, then check compiler and actual caller compatibility. Stop on a nonphysical credit or an unfavorable complete relaxation.",
+    "bottleneck_link": "Judge the actual current supplier contract and complete signed target/paid bill; frozen-profile ceilings require an applicability proof.",
+    "allowed_variations": [
+      "One exact source head, helper/cut region or quantified proposition and changed hypothesis; check live handoffs first. Different seeds are not different scopes."
+    ],
+    "scope_change_triggers": [
+      "Changing source head, helper/cut domain, operation alphabet or caller boundary without recording the new scope.",
+      "Treating an archived proposal or local numerical pass as complete supplier admission."
+    ],
+    "overlap_terms": [
+      "target credit",
+      "helper split",
+      "coverage law"
+    ],
+    "existing_checks": [
+      {
+        "scope": "Selected PR5599: complete helper-subspace construction and independent caller replay; five framework assumptions remain.",
+        "evidence": "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md",
+        "remaining": "The next experiment must identify uncovered source-specific work. Successor archive intake does not transfer this admission.",
+        "status": "reviewed",
+        "source_commit": "09dd0965711ca7eeb169057a420abab1c9052137",
+        "selected_record_sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5"
+      }
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Prove power-only collapse of every complete fixed-cardinality layer in the pinned antichain family.",
+  "scope_change_reason": "The lateral-thinking assignment asks for escape proofs and failed constructions. This is an accounting side theorem answering a predecessor open question, not a new productive target.",
+  "bottleneck_link": "Distinguishes power-moment search difficulty from all-concave frontier size; physical applicability remains open.",
+  "budget_minutes": 15,
+  "stop_rule": "Stop after the uniform exchange proof and exact rational corroboration; no physical search.",
+  "outcome": "completed",
+  "evidence": "README.md proof and verify.py exact intervals, hinge and boundary controls.",
+  "author": "zxv (https://github.com/zxv), with substantial OpenAI Codex assistance",
+  "ai_assistance": "Substantial OpenAI Codex assistance in derivation, implementation and writing.",
+  "open_questions": [
+    {
+      "question": "Can a legal supplier realize two independent equal-width exchanges?",
+      "why_it_matters": "Would apply the accounting reduction to physical kappa work.",
+      "first_step": "Extract two source-bound paid exchanges and complete boundary/fee ledger.",
+      "blocked_by": "No physical realization in the synthetic family."
+    },
+    {
+      "question": "What is the power frontier of an incomplete layer?",
+      "why_it_matters": "The canonical minimum may be absent from an actual candidate pool.",
+      "first_step": "Certify one pair with index-incomparable sorted subsets over 0<p<1.",
+      "blocked_by": "The exchange theorem alone does not compare these pairs."
+    }
+  ]
+}

research/power-layer-collapse/VERIFY.json

@@ -0,0 +1,32 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/concave-invoice-antichains/README.md"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Exact rational root intervals, every complete layer through m=11, hinge/endpoints/reversed-domain/changed-contract controls."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-O",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Same checks with Python assertions disabled."
+    }
+  ]
+}